Finite Group Databasebeta

← C31×#19051 #21475810

Subgroups — C31×#19051 #21475810

Order 7936.

950 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3545C211yesmaximal, characteristic
2C31×#3238C212yesmaximal, complemented
3C2×C31×#497C211yesmaximal, characteristic, complemented
4C31×#1661C214yesmaximal, complemented
5C31×#1660C212yesmaximal, complemented
6C31×#1659C212yesmaximal, complemented
7C31×#1533C211yesmaximal, characteristic, complemented
8C31×#1532C212yesmaximal, complemented
9C22×Q16×C31—21no—
10C22×SD16×C31—21no—
11C31×#563C2211yescharacteristic
12C31×#562C2212yes—
13C31×H(3,𝔽4)C2211yescharacteristic
14C31×#555—21no—
15C31×#551—21no—
16C31×#551—22no—
17C31×#546—22no—
18C31×#542—22no—
19C31×#538—22no—
20C31×#536—21no—
21C31×#534—21no—
22C2×C31×C4⋊Q8C2211yescharacteristic, complemented
23C31×#529C2212yes—
24C2×C31×#124C2211yescharacteristic
25C31×#501C2214yescomplemented
26C31×#497—21no—
27C31×#497—21no—
28C31×#497C2212yes—
29C31×#497C2212yescomplemented
30C31×#489C2214yescomplemented
31C31×#483—22no—
32C31×#480—22no—
33C31×#477—22no—
34C31×#476—22no—
35C31×#432C2212yescomplemented
36C2×C31×#97C2211yescharacteristic, complemented
37C31×#330C2214yescomplemented
38C31×#329C2212yes—
39C31×#329C2214yescomplemented
40C31×#329C2212yescomplemented
41C2×#41×C31—21no—
42C22×Q8×C31—21no—
43C22×Q8×C31D411yescharacteristic
44C22×D4×C31—21no—
45C22×D4×C31D411yescharacteristic
46C2×Q16×C31—22no—
47C2×Q16×C31—22no—
48C2×Q16×C31—44no—
49C2×SD16×C31—22no—
50C2×SD16×C31—22no—
51C2×SD16×C31—44no—
52C22×C8×C31—21no—
53C31×C4⋊Q8—21no—
54C31×C4⋊Q8—21no—
55C31×C4⋊Q8—22no—
56C31×C4⋊Q8—21no—
57C31×C4⋊Q8C2312yes—
58C31×#126—22no—
59C31×#126—22no—
60C31×#126—22no—
61C31×#125C2312yes—
62C31×#124—21no—
63C31×#124—22no—
64C31×#124—21no—
65C31×#124—22no—
66C31×#124—22no—
67C31×#124D411yescharacteristic
68C31×#124D411yescharacteristic
69C31×#124C2312yes—
70C31×#124C2311yescharacteristic
71C31×#124C2311yescharacteristic
72C31×#123—21no—
73C31×#123—22no—
74C31×#123—22no—
75C31×#123—21no—
76C31×#123—22no—
77C31×#123—22no—
78C31×#123—22no—
79C31×C22⋊Q8—22no—
80C31×C22⋊Q8—22no—
81C31×C22⋊Q8—22no—
82C31×C22⋊Q8—22no—
83C31×C22⋊Q8D412yes—
84C31×#121—22no—
85C31×#121—22no—
86C31×#121—22no—
87C31×#121—22no—
88C31×#121D412yes—
89C31×#120—21no—
90C31×#120—22no—
91C31×#120—21no—
92C31×#120—22no—
93C4×Q8×C31D412yes—
94C4×D4×C31—22no—
95C4×D4×C31—22no—
96C4×D4×C31—22no—
97C4×D4×C31D412yes—
98C31×#117—21no—
99C31×#117—22no—
100C31×#117—22no—
101C2×C4⋊C4×C31—22no—
102C2×C22⋊C4×C31—21no—
103C2×C22⋊C4×C31—22no—
104C2×C22⋊C4×C31—22no—
105C2×C42×C31C2311yescharacteristic
106C31×#107—24no—
107C31×#106—24no—
108C31×C4⋊C8—22no—
109C31×#103—24no—
110C31×#102—24no—
111C31×C22⋊C8—22no—
112C31×#97—21no—
113C31×#97C2312yes—
114C31×#97C2314yes—
115C24×C31—21no—
116C24×C31—21no—
117#41×C31—82no—
118C2×Q8×C31—21no—
119C2×Q8×C31—21no—
120C2×Q8×C31—22no—
121C2×Q8×C31—42no—
122C2×Q8×C31—42no—
123C2×Q8×C31—42no—
124C2×Q8×C31—41no—
125C2×Q8×C31C2×D412yes—
126C2×Q8×C31C2×D411yescharacteristic
127C2×Q8×C31C2×D411yescharacteristic
128C2×D4×C31—22no—
129C2×D4×C31—22no—
130C2×D4×C31—22no—
131C2×D4×C31—21no—
132C2×D4×C31—22no—
133C2×D4×C31—21no—
134C2×D4×C31—42no—
135C2×D4×C31—82no—
136C2×D4×C31C2×D412yes—
137C2×D4×C31C2×D411yescharacteristic
138C2×D4×C31C2×D411yescharacteristic
139C22×C4×C31—21no—
140C22×C4×C31—22no—
141C22×C4×C31—22no—
142C22×C4×C31—21no—
143C22×C4×C31—22no—
144C22×C4×C31—41no—
145C22×C4×C31C2×D411yescharacteristic
146C22×C4×C31C2×D412yes—
147Q16×C31—44no—
148Q16×C31—44no—
149SD16×C31—44no—
150SD16×C31—44no—
151C2×C8×C31—24no—
152C2×C8×C31—22no—
153C2×C8×C31—24no—
154C4⋊C4×C31—22no—
155C4⋊C4×C31—22no—
156C4⋊C4×C31—22no—
157C4⋊C4×C31—22no—
158C4⋊C4×C31—22no—
159C4⋊C4×C31—22no—
160C4⋊C4×C31—22no—
161C4⋊C4×C31—24no—
162C4⋊C4×C31—42no—
163C4⋊C4×C31C2×D412yes—
164C4⋊C4×C31C2×D412yes—
165C22⋊C4×C31—22no—
166C22⋊C4×C31—21no—
167C22⋊C4×C31—21no—
168C22⋊C4×C31—22no—
169C22⋊C4×C31—22no—
170C22⋊C4×C31—22no—
171C22⋊C4×C31—22no—
172C22⋊C4×C31—22no—
173C22⋊C4×C31—22no—
174C22⋊C4×C31—22no—
175C22⋊C4×C31—22no—
176C22⋊C4×C31—22no—
177C22⋊C4×C31—22no—
178C22⋊C4×C31—22no—
179C22⋊C4×C31—22no—
180C42×C31—21no—
181C42×C31—21no—
182C42×C31—22no—
183C42×C31C2×D411yescharacteristic
184C42×C31C2×D412yes—
185C42×C31C2411yescharacteristic
186#19051C3111yesmaximal, characteristic, direct product factor, Sylow
187C23×C31—21no—
188C23×C31—22no—
189C23×C31—21no—
190C23×C31—42no—
191C23×C31—82no—
192C23×C31—81no—
193C23×C31—82no—
194C23×C31#12011yescharacteristic
195Q8×C31—41no—
196Q8×C31—41no—
197Q8×C31—42no—
198Q8×C31—42no—
199Q8×C31—42no—
200Q8×C31—81no—
201Q8×C31—82no—
202Q8×C31—81no—
203D4×C31—41no—
204D4×C31—41no—
205D4×C31—42no—
206D4×C31—44no—
207D4×C31—81no—
208D4×C31—82no—
209D4×C31—82no—
210D4×C31—82no—
211D4×C31—81no—
212C2×C4×C31—22no—
213C2×C4×C31—22no—
214C2×C4×C31—21no—
215C2×C4×C31—22no—
216C2×C4×C31—22no—
217C2×C4×C31—21no—
218C2×C4×C31—42no—
219C2×C4×C31—42no—
220C2×C4×C31—42no—
221C2×C4×C31—82no—
222C2×C4×C31—82no—
223C2×C4×C31—81no—
224C2×C4×C31—82no—
225C2×C4×C31—82no—
226C2×C4×C31#12012yes—
227C2×C4×C31#12011yescharacteristic
228C2×C4×C31C22×D412yes—
229C2×C4×C31C22×D411yescharacteristic
230C8×C31—24no—
231C8×C31—44no—
232#3545C2×C3111yescharacteristic
233#3238C2×C3112yescomplemented
234C2×#497C2×C3111yescharacteristic, complemented
235#1661C2×C3114yescomplemented
236#1660C2×C3112yescomplemented
237#1659C2×C3112yescomplemented
238#1533C2×C3111yescharacteristic, complemented
239#1532C2×C3112yescomplemented
240C22×C31—22no—
241C22×C31—42no—
242C22×C31—41no—
243C22×C31—41no—
244C22×C31—42no—
245C22×C31—42no—
246C22×C31—42no—
247C22×C31—42no—
248C22×C31—162no—
249C22×C31—162no—
250C22×C31#45312yes—
251C22×C31C2×#12011yescharacteristic
252C4×C31—21no—
253C4×C31—21no—
254C4×C31—22no—
255C4×C31—22no—
256C4×C31—41no—
257C4×C31—41no—
258C4×C31—42no—
259C4×C31—42no—
260C4×C31—42no—
261C4×C31—82no—
262C22×Q16—21no—
263C22×SD16—21no—
264#563C22×C3111yescharacteristic
265#562C22×C3112yes—
266H(3,𝔽4)C22×C3111yescharacteristic
267#555—21no—
268#551—21no—
269#551—22no—
270#546—22no—
271#542—22no—
272#538—22no—
273#536—21no—
274#534—21no—
275C2×C4⋊Q8C22×C3111yescharacteristic, complemented
276#529C22×C3112yes—
277C2×#124C22×C3111yescharacteristic
278#501C22×C3114yescomplemented
279#497—21no—
280#497—21no—
281#497C22×C3112yes—
282#497C22×C3112yescomplemented
283#489C22×C3114yescomplemented
284#483—22no—
285#480—22no—
286#477—22no—
287#476—22no—
288#432C22×C3112yescomplemented
289C2×#97C22×C3111yescharacteristic, complemented
290#330C22×C3114yescomplemented
291#329C22×C3112yes—
292#329C22×C3114yescomplemented
293#329C22×C3112yescomplemented
294C2×C31—22no—
295C2×C31—82no—
296C2×C31—81no—
297C2×C31—81no—
298C2×C31C2×#45311yescharacteristic
299C2×C31#302212yes—
300C2×#41—21no—
301C22×Q8—21no—
302C22×Q8D4×C3111yescharacteristic
303C22×D4—21no—
304C22×D4D4×C3111yescharacteristic
305C2×Q16—22no—
306C2×Q16—22no—
307C2×Q16—44no—
308C2×SD16—22no—
309C2×SD16—22no—
310C2×SD16—44no—
311C22×C8—21no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—22no—
315C4⋊Q8—21no—
316C4⋊Q8C23×C3112yes—
317#126—22no—
318#126—22no—
319#126—22no—
320#125C23×C3112yes—
321#124—21no—
322#124—22no—
323#124—21no—
324#124—22no—
325#124—22no—
326#124D4×C3111yescharacteristic
327#124D4×C3111yescharacteristic
328#124C23×C3111yescharacteristic
329#124C23×C3111yescharacteristic
330#124C23×C3112yes—
331#123—21no—
332#123—22no—
333#123—22no—
334#123—21no—
335#123—22no—
336#123—22no—
337#123—22no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—22no—
342C22⋊Q8D4×C3112yes—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—22no—
347#121D4×C3112yes—
348#120—21no—
349#120—22no—
350#120—21no—
351#120—22no—
352C4×Q8D4×C3112yes—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4D4×C3112yes—
357#117—21no—
358#117—22no—
359#117—22no—
360C2×C4⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C42C23×C3111yescharacteristic
365#107—24no—
366#106—24no—
367C4⋊C8—22no—
368#103—24no—
369#102—24no—
370C22⋊C8—22no—
371#97—21no—
372#97C23×C3114yes—
373#97C23×C3112yes—
374C31#1905111yescharacteristic, direct product factor, Sylow
375C24—21no—
376C24—21no—
377#41—82no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—41no—
385C2×Q8C2×D4×C3112yes—
386C2×Q8C2×D4×C3111yescharacteristic
387C2×Q8C2×D4×C3111yescharacteristic
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—21no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4C2×D4×C3112yes—
397C2×D4C2×D4×C3111yescharacteristic
398C2×D4C2×D4×C3111yescharacteristic
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4C2×D4×C3111yescharacteristic
406C22×C4C2×D4×C3112yes—
407Q16—44no—
408Q16—44no—
409SD16—44no—
410SD16—44no—
411C2×C8—24no—
412C2×C8—22no—
413C2×C8—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4C2×D4×C3112yes—
424C4⋊C4C2×D4×C3112yes—
425C22⋊C4—22no—
426C22⋊C4—21no—
427C22⋊C4—21no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C42—21no—
441C42—21no—
442C42—22no—
443C42C2×D4×C3111yescharacteristic
444C42C2×D4×C3112yes—
445C42C24×C3111yescharacteristic
446C23—21no—
447C23—22no—
448C23—21no—
449C23—42no—
450C23—82no—
451C23—81no—
452C23—82no—
453C23C31×#12011yescharacteristic
454Q8—41no—
455Q8—41no—
456Q8—42no—
457Q8—42no—
458Q8—42no—
459Q8—81no—
460Q8—82no—
461Q8—81no—
462D4—41no—
463D4—41no—
464D4—42no—
465D4—44no—
466D4—81no—
467D4—82no—
468D4—82no—
469D4—82no—
470D4—81no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C31×#12012yes—
486C2×C4C31×#12011yescharacteristic
487C2×C4C22×D4×C3112yes—
488C2×C4C22×D4×C3111yescharacteristic
489C8—24no—
490C8—44no—
491C22—22no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—162no—
500C22—162no—
501C22C31×#45312yes—
502C22C2×C31×#12011yescharacteristic
503C4—21no—
504C4—21no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—41no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C2—22no—
514C2—81no—
515C2—82no—
516C2—81no—
517C2C2×C31×#45311yescharacteristic
518C2C31×#302212yes—
subgroupG/H
center, socleC22×C31C2×#120
commutator, Frattini, cosocleC42C24×C31
prime radicalC31×#3545C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C31×#3238C2semidirect
3. C2×C31×#497C2semidirect
4. C31×#1661C2semidirect
5. C31×#1660C2semidirect
6. C31×#1659C2semidirect
7. C31×#1533C2semidirect
8. C31×#1532C2semidirect
22. C2×C31×C4⋊Q8C22semidirect
25. C31×#501C22semidirect
29. C31×#497C22semidirect
30. C31×#489C22semidirect
35. C31×#432C22semidirect
36. C2×C31×#97C22semidirect
37. C31×#330C22semidirect
39. C31×#329C22semidirect
40. C31×#329C22semidirect
186. #19051C31direct (N × H)
233. #3238C2×C31semidirect
234. C2×#497C2×C31semidirect
235. #1661C2×C31semidirect
236. #1660C2×C31semidirect
237. #1659C2×C31semidirect
238. #1533C2×C31semidirect
239. #1532C2×C31semidirect
275. C2×C4⋊Q8C22×C31semidirect
278. #501C22×C31semidirect
282. #497C22×C31semidirect
283. #489C22×C31semidirect
288. #432C22×C31semidirect
289. C2×#97C22×C31semidirect
290. #330C22×C31semidirect
292. #329C22×C31semidirect
293. #329C22×C31semidirect
374. C31#19051direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#35452
C31×#32382
C2×C31×#4972
C31×#16612
C31×#16602
C31×#16592
C31×#15332
C31×#15322
#1905131